Also in this extended model, considering only the Planck satellite data on the CMB temperature anisotropies and polarization, we obtain a preference for a curved cosmological spacetime at more than 95% CL, with the constraint on the curvature parameter reading \(\Omega _{K0}=-0.073^{+0.060}_{-0.071}\)  at 95% CL. As for the previous baseline case, this preference is strongly reduced when the Planck data are combined with the BAO datasets that are in tension with it, and their combination prefers spatial flatness within one standard deviation. As concerns the value of the expansion rate today, due to its degeneracy with the mass of the active neutrinos, in this extended parameter space the CMB measurements prefer further lower values of \(H_0\)  giving a less tight bound \(H_0=48.3^{+5.7}_{-5.9}\)  km/s/Mpc at 68% CL, which is also in strong tension with the SH0ES independent measurement of \(H_0\)  [1]}, at the level of \(\sim 4.3\sigma \) . Combining the CMB measurements with the BAO and Pantheon data, the tension between the different datasets remains statistically significant. In particular the values inferred combining CMB and BAO, \(H_0=67.84\pm 0.67\)  km/s/Mpc at 68% CL, is similar to the result obtained in the flat \(\Lambda \) CDM model, while combining CMB and Pantheon we get \(H_0=65.3^{+2.2}_{-2.3}\)  km/s/Mpc at 68% CL. Notice that both these values are in tension with the SH0ES collaboration at the level of \(4.3\sigma \)  and \(3.2\sigma \) , respectively. Furthermore, due to the strong anti-correlation between \(H_0\)  and the matter density parameter, lower values of \(H_0\)  prefer higher values of \(\Omega _{m0}\) , resulting in a severe tension also for the \(S_8\)  parameter. Comparing the bound of \(S_8\)  derived by the CMB measurements alone with respect to the constraint obtained including also the BAO and Pantheon data, we can observe a tension at the level of about \(\sim 3\sigma \) .
Finally, concerning the neutrino masses, the CMB data alone constrain their total mass to be \(M_{\nu }<0.79\)  eV at 95% CL, i.e. a factor more 3 relaxed than the flat \(\Lambda \) CDM+\(M_{\nu }\)  scenario
However, neutrinos suppress structure formation at small scales and therefore astrophysical galaxy clustering measurements turn out to be crucial to improve the constraining power on their total mass. For this reason, the upper bounds on \(M_{\nu }\)  can be significantly improved to \(M_{\nu }<0.17\)  eV (always at 95% CL) including also the Baryon Acoustic Oscillation and the Pantheon datasets. This upper limit should be compared with that obtained for a flat \(\Lambda \) CDM+\(M_{\nu }\)  scenario, that is \(M_{\nu }<0.13\)  eV at 95% CL for CMB+BAO. In other words, the assumption of flatness produces a much stronger upper limit on the total neutrino mass than when the curvature can vary, biasing conclusions that are important to laboratory experiments.
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